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In this paper, the third of its series, we prove that the sobolev spaces ofL^p_k approximate solutions to the Dirichlet problem for the epsilon-Yang Millsequations on a four dimensional disk, carry a natural manifold structure (moreprecisely a natural structure of Banach bundle), for p(k+1)> 4. All resultsapply also if the four-dimensional disk is replaced by a general compactmanifold with boundary, and SU(2) is replaced by any compact Lie group. We alsoconstruct bases for the tangent space to the space of approximate solutions,thus showing that this space is 8-dimensional for epsilon sufficiently small,and prove some technical results used in Parts I and II for the proof of theexistence of multiple solution and, in particular, non-minimal ones, for thisnon-compact variational problem.

Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part III